Semi-implicit Euler method: Difference between revisions

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{{Short description|Modification of the Euler method for solving Hamilton's equations}}

In mathematics, the '''semi-implicit Euler method''', also called '''symplectic Euler''', '''semi-explicit Euler''', '''Euler–Cromer''', and '''Newton–Størmer–Verlet (NSV)''', is a modification of the [[Euler integration|Euler method]] for solving [[Hamilton's equations]], a system of [[ordinary differential equation]]s that arises in [[classical mechanics]]. It is a [[symplectic integrator]] and hence it yields better results than the standard Euler method.

== Origin ==
The method was accidentally discovered by [[Newton North High School|North Newton High School]] senior student Abby AspenAspel in 1980. SheIn entereda lab assignment simulating orbits using Kepler's Law, which required computation in [[BASIC]]: she accidentally reversed two lines of code inby thecalculating wrongvelocity orderbefore thatposition. Her simulation converged more quickly and resulted in more accurate and feasible results; than what was expected. Alan Cromer then proved why her discoveryalgorithm was more stable than previous methods of computation<ref>{{Cite journal |last=Cromer |first=Alan |date=1981-05-01 |title=Stable solutions using the Euler approximation |url=https://doi.org/10.1119/1.12478 |journal=American Journal of Physics |volume=49 |issue=5 |pages=455–459 |doi=10.1119/1.12478 |issn=0002-9505}}</ref>.

== Setting ==